solenoid acting up

[Michael]

How do you test a test solenoid? 

my A/T switch does not completely shut off the power to the dash 

I got out my meter and performed these tests;

#1 found it had 3.8 to 4.5 volts still passing through the relay when switched off and went to "0" only when I activated another switch with a larger draw like the (mirror heat) The step extend switch just changes the tone of a low voltage chatter. 

#2 with load wire disconnected from the switch, it seemed to work consistently, showing only .01 or.02 VDC on the meter.

I'm not sure what that means or how I should really test it.... why would it work without a load? 

I've checked temperature of the relay when left on and gets up to 120+ degrees. Is that normal? I hate to lose all that energy to heat. 

Thanks for any help